Revisiting Mayer: Symmetric solutions for sporadic cases of the Map Color Theorem

TL;DR

This paper revisits Mayer's solutions for specific cases of the Map Color Theorem, providing new embeddings for K18 and K23 using current graph theory, and improving understanding of these sporadic cases.

Contribution

It offers a new interpretation of Mayer's solutions through current graph theory and presents improved embeddings for K18 and K23.

Findings

New embeddings for K18 and K23 graphs.

Interpretation of Mayer's solutions via current graphs.

Enhanced understanding of sporadic cases in the Map Color Theorem.

Abstract

The original proof of the genus of the complete graphs $K_n$ depended on Mayer's \emph{ad hoc} solutions for $n = 18, 20, 23$. Recently, an improved solution for $K_{20}$ was found by the author. The purpose of this note is to use the theory of current graphs to interpret the aforementioned result and to provide new embeddings of $K_{18}$ and $K_{23}$.